In order to achieve the climate targets, a mix of different powertrain technologies must be pursued to effectively reduce emissions. By producing hydrogen based on renewable energy sources, it becomes a reasonable choice for fueling internal combustion engines. The specific molecular properties of hydrogen thereby open up new possibilities for favorably influencing the combustion process of engines. The present paper deals with the analysis of a single-cylinder engine with passive pre-chamber ignition and a port fuel injection system, which was adapted for lean hydrogen operation. In this way, the test unit was operated in various load and speed ranges with lambda values from 1.5 to 2.5 and achieved up to 23 bar indicated mean effective pressure. The focus of this work is on the numerical investigation of the hydrogen combustion and its effects on the engine system. Special attention is hereby paid to the influence of different lambda operations. Simulations were carried out to evaluate the heat transfer towards the cooling system and to determine energy losses dependent on the gas temperatures. The validated 3D-CFD simulation illustrates the thermodynamic properties, as well as the interaction of injection strategies and mixture formation inside the cylinder and pre-chamber. The analysis points out that lean operation across all loads is advantageous in terms of indicated efficiency and particularly in lower loads up to 6 %-pts can be achieved by applying a de-throttling strategy. By going beyond test bench limitations, the virtual environment shows that the engine is knock-limited with lambda 1.5 and increased combustion temperatures imply high NOx emissions at high loads. In contrast, with lambda 2.5, the pressure gradient and turbulence level decrease sharply, so that a boost pressure of 4.5 bar is required to achieve the maximum load with a peak cylinder pressure of 180 bar.